一元一次方程X+X/(1+2)+X/(1+2+3)+.X/(1+2+3+.2009)=2009的解我想知道解题思路

Question

一元一次方程X+X/(1+2)+X/(1+2+3)+.X/(1+2+3+.2009)=2009的解
我想知道解题思路

Answer 1

X+X/(1+2)+X/(1+2+3)+.X/(1+2+3+.2009)=2009
x+2x/[2(1+2)+2x/[3(1+3)]+······+2x/[2009(1+2009)]=2009
2x/(1×2)+2x/(2×3)+2x/(3×4)+······+2x/(2009×2010)=2009
x/(1×2)+x/(2×3)+x/(3×4)+······+x/(2009×2010)=2009/2
x/1-x/2+x/2-x/3+x/3-x/4+······x/2009-x/2010=2009/2
x-x/2010=2009/2
2019x/2010=2009/2
x=2010/2
x=1005